Why did Leo Higgins have a burnt nose in Muriel's Wedding?

The character to whom you are referring was named Leo Higgins (Rob Steele). The character of Coach Ken Blundell (Chris Haywood), is actually evenly tanned, as far as the viewer can tell. Leo Higgins is the man whom approaches Muriel at Tanya's wedding with the burned nose.

I always thought that this was, at least in part, referencing the character of Bardolph, who appears in several of William Shakespeare's plays. Like Leo, he is always a minor character; additionally, like him, has severe skin issues on his nose ("his face is Lucifer’s privy kitchen, where he doth nothing but roast malt-worms"). Though really used for comic relief, Shakespeare also reflects Henry's rise from Prince to King via Bardolph, of which indeed — as Leo supported Bill Heslop's political motives — he 'supported.'

It's also worth mentioning, as there are many parallels between the two — that Bardolph was seemingly always trying to convince Henry to commit 'petty crimes.' Whether or not we are meant to assume Leo had anything to do with Bill's bribery woes is unclear but Bill did reward loyalty, especially from those he deemed successful.

Leo did say the scarring was due to sunburn; this reference is correct, save for the character name.

As far as I can tell, Ken is a swimming coach. It means that he is spending a lot of time outside, near the pool and inside of it, causing his nose to get burnt. His whole body is one giant sunburn, actually, only the nose starts peeling the fastest. Ken doesn't strike me as a person who would use sunscreen but would actually avoid using it, because he is "manly man". Could also signify a person "in hot water", because Ken is forced by external circumstances to defend Bill. It's been years since I watched this movie, so I might be way off.